Not seeing the site, I would be afraid to say. However, if the stones are the only support, the basement would probably move downward, unless it's also supported in the middle by stones.
If I tackled such a job, and I knew the basement was soundly built (either concrete or block masonry) I would dig out one end and place concrete blocks - ground depth up to basement-line (which I would leave during the pour), then dig mid-way and do the same, as well as the next corner. When one side was completed, I would tackle the opposite side, and then the remaining end walls.
Everything depends on how well the basement is constructed. If the floor is thick, re-bar enforced then you might not have a problem a section at a time. Good luck. ---Jack